extension field
Noun
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Noun
extension field (plural extension fields)
- (algebra, field theory) A field L which contains a subfield K, called the base field, from which it is generated by adjoining extra elements.
- 1992, James Oxley, Matroid Theory, Oxford University Press, 2006, Paperback, page 215 ↗,
- Suppose F is a subfield of the field K. Then K is called an extension field of F. So, for instance, GF(4) and GF(8) are extension fields of GF(2), although GF(8) is not an extension field of GF(4).
- 1995, Terence Jackson, From Polynomials to Sums of Squares, Taylor & Francis, page 56 ↗,
- This extension field of F always contains a root of f in the sense that if K = F[x]/(f(x)) then x is a root of f(y) in K[y]. It then follows that any polynomial will have roots, either in the original field of its coefficients or in some extension field.
- 1998, Neal Koblitz, Algebraic Aspects of Cryptography, Volume 3, Springer, page 53 ↗,
- An extension field, by which we mean a bigger field containing F, is automatically a vector space over F. We call it a finite extension if it is a finite vector space. By the degree of a finite extension we mean its dimension as a vector space. One common way of obtaining extension fields is to adjoin an element to F: we say that K = F(\alpha) if K is the field consisting of all rational expressions formed using \alpha and elements of F.
- 1992, James Oxley, Matroid Theory, Oxford University Press, 2006, Paperback, page 215 ↗,
- (field that contains a subfield) extension (where the base field is given)
This text is extracted from the Wiktionary and it is available under the CC BY-SA 3.0 license | Terms and conditions | Privacy policy 0.002